Robust Optimal Designs when Missing Data Happen at Random

TL;DR

This paper develops a robust optimal design framework for predicting responses in regression models with missing data, aiming to minimize worst-case prediction errors under model misspecification and missing at random conditions.

Contribution

It introduces a minimax design approach that accounts for model uncertainty and missing data, providing robust designs that protect against increased prediction errors.

Findings

Derived analytical expressions for the worst-case mean squared prediction errors.

Proposed a minimax optimization method for robust design selection.

Demonstrated robustness of the designs through theoretical analysis.

Abstract

In this article, we investigate the robust optimal design problem for the prediction of response when the fitted regression models are only approximately specified, and observations might be missing completely at random. The intuitive idea is as follows: We assume that data are missing at random, and the complete case analysis is applied. To account for the occurrence of missing data, the design criterion we choose is the mean, for the missing indicator, of the averaged (over the design space) mean squared errors of the predictions. To describe the uncertainty in the specification of the real underlying model, we impose a neighborhood structure on the deterministic part of the regression response and maximize, analytically, the \textbf{M}ean of the averaged \textbf{M}ean squared \textbf{P}rediction \textbf{E}rrors (MMPE), over the entire neighborhood. The maximized MMPE is the ``worst'' loss in the neighborhood of the fitted regression model. Minimizing the maximum MMPE over the class of designs, we obtain robust ``minimax'' designs. The robust designs constructed afford protection from increases in prediction errors resulting from model misspecifications.